U.S. patent application number 12/450746 was filed with the patent office on 2010-02-25 for method and apparatus for selectively reducing noise in a digital signal.
This patent application is currently assigned to THOMSON LICENSING. Invention is credited to Malte Borsum, Joern Jachalsky.
Application Number | 20100049778 12/450746 |
Document ID | / |
Family ID | 39227018 |
Filed Date | 2010-02-25 |
United States Patent
Application |
20100049778 |
Kind Code |
A1 |
Jachalsky; Joern ; et
al. |
February 25, 2010 |
METHOD AND APPARATUS FOR SELECTIVELY REDUCING NOISE IN A DIGITAL
SIGNAL
Abstract
Wavelet thresholding using discrete wavelet transforms is a
sophisticated and effective approach for noise reduction. However,
usage of integer arithmetic implies that not the full range of
input values can be used. A method for selectively reducing noise
in a digital signal having a first range of values comprises steps
of decomposing the digital signal to a plurality of frequency
sub-bands, wherein before, during or after the decomposing the
digital signal or at least one sub-band is expanded by one or more
bits to a second range of integer values, removing in at least one
of the frequency sub-bands values that are below a threshold,
re-combining the frequency sub-bands, after removing said values
that are below a threshold, into an expanded output signal, and
de-expanding the expanded output signal, wherein a signal having
the first range of values is obtained.
Inventors: |
Jachalsky; Joern; (Hannover,
DE) ; Borsum; Malte; (Hannover, DE) |
Correspondence
Address: |
Robert D. Shedd, Patent Operations;THOMSON Licensing LLC
P.O. Box 5312
Princeton
NJ
08543-5312
US
|
Assignee: |
THOMSON LICENSING
Boulogne-Billancourt
FR
|
Family ID: |
39227018 |
Appl. No.: |
12/450746 |
Filed: |
April 8, 2008 |
PCT Filed: |
April 8, 2008 |
PCT NO: |
PCT/EP2008/054221 |
371 Date: |
October 9, 2009 |
Current U.S.
Class: |
708/400 |
Current CPC
Class: |
G06F 17/148 20130101;
G06T 5/002 20130101; G06T 5/10 20130101; G06T 2207/20064
20130101 |
Class at
Publication: |
708/400 |
International
Class: |
G06F 17/14 20060101
G06F017/14 |
Foreign Application Data
Date |
Code |
Application Number |
Apr 12, 2007 |
EP |
07106083.4 |
Claims
1-12. (canceled)
13. A method for selectively reducing noise in a digital signal
having a first range of values, comprising the steps of decomposing
the digital signal to a plurality of frequency sub-bands using
integer arithmetic and wavelet transform, and expanding the digital
signal or at least one of said frequency sub-bands by one or more
bits before, during or after the step of decomposing, wherein at
least one expanded frequency sub-band having a higher range of
integer values than the first range of integer values is obtained,
and wherein said expanding is achieved either by adding one or more
bits at the LSB position and setting the added bit or bits to a
value that is obtained from said digital signal, or by omitting a
division operation for the at least one frequency sub-band during
said step of decomposing; reducing or removing in at least one
expanded frequency sub-band values that are below a threshold; and
reconstructing from the frequency sub-bands, after said step of
reducing or removing in the at least one expanded frequency
sub-band the values that are below a threshold, an output signal
having the first range of values, wherein the at least one expanded
frequency sub-band or the reconstructed output signal is
de-expanded and wherein integer arithmetic is used.
14. Method according to claim 13, wherein separate thresholds are
applied to more than one frequency sub-band, and wherein the
threshold of each frequency sub-band can differ from the thresholds
of other frequency sub-bands.
15. Method according to claim 13, wherein the one or more
thresholds are integer thresholds.
16. Method according to claim 13, wherein, in the case of said
expanding being achieved by omitting a division operation, the one
or more expanding bits are added at the LSB position and are set to
zero, and the step of de-expanding comprises rounding to the
nearest integer value.
17. Method according to claim 13, wherein, in the case of said
expanding being achieved by adding one or more bits at the LSB
position, the value that is obtained from said digital signal is
obtained from the most significant bits (MSBs) of said digital
signal.
18. Method according to claim 13, wherein said step of de-expanding
the expanded output signal comprises removing the least significant
bits.
19. Method according to claim 13, wherein the step of decomposing
is performed by a digital wavelet transform and the step of
reconstructing is performed by an inverse digital wavelet
transform.
20. Apparatus for selectively reducing noise in a digital signal
having a first range of values, comprising means for decomposing
the digital signal to a plurality of frequency sub-bands using
integer arithmetic and wavelet transform, and means for expanding
the digital signal or at least one of said frequency sub-bands by
one or more bits before, during or after the step of decomposing,
wherein at least one expanded frequency sub-band having a higher
range of integer values than the first range of integer values is
obtained, and wherein said expanding is achieved either by adding
one or more bits at the LSB position and setting the added bit or
bits to a value that is obtained from said digital signal, or by
omitting a division operation for the at least one frequency
sub-band during said decomposing; means for reducing or removing in
at least one expanded frequency sub-band values that are below a
threshold; and means for reconstructing from the frequency
sub-bands, including the at least one expanded frequency sub-band,
an output signal having the first range of values, wherein the
apparatus comprises means for de-expanding the at least one
expanded frequency sub-band or the reconstructed output signal and
wherein integer arithmetic is used.
21. Apparatus according to claim 20, wherein the means for
expanding the digital signal further has means for adding one or
more bits at the LSB position; means for obtaining one or more bits
of said digital signal; and means for setting the added one or more
bits to the obtained one or more bits of said digital signal.
22. Apparatus according to claim 20, wherein, in the case of said
expanding being achieved by adding one or more bits at the LSB
position, the added value that is obtained from said digital signal
is obtained from the MSBs of said digital signal.
23. Apparatus according to claim 20, wherein said means for
de-expanding the expanded output signal has means for removing the
LSBs.
24. Method according to claim 13, wherein said expanding is
achieved by omitting a division operation for the at least one
frequency sub-band during said step of decomposing, and wherein
said decomposing is performed according to
y(2n+1)=2*x(2n+1)-(x(2n)+x(2n+2)) and/or
y(2n)=8*x(2n)+y(2n-1)+y(2n+1), and wherein the expansion for the
lowest frequency sub-band can be selected to be 6, 3 or 0 bit.
25. Method according to claim 24, wherein said decomposing is
performed into four sub-bands LL, HL, LH and HH, and wherein the
expansion is 4, 1 or 0 bit for the sub-band HL, or the expansion is
4, 3 or 0 bit for sub-band LH, or the expansion is 2, 1 or 0 bit
for sub-band HH.
26. Apparatus according to claim 20, wherein said expanding is
achieved by omitting a division operation for the at least one
frequency sub-band during said step of decomposing, and wherein
said decomposing is performed according to
y(2n+1)=2*x(2n+1)-(x(2n)+x(2n+2)) and/or
y(2n)=8*x(2n)+y(2n-1)+y(2n+1), and wherein the expansion for the
lowest frequency sub-band can be selected to be 6, 3 or 0 bit.
27. Apparatus according to claim 26, wherein said decomposing is
performed into four sub-bands LL, HL, LH and HH, and wherein the
expansion is 4, 1 or 0 bit for the sub-band HL, or the expansion is
4, 3 or 0 bit for sub-band LH, or the expansion is 2, 1 or 0 bit
for sub-band HH.
Description
FIELD OF THE INVENTION
[0001] This invention relates to a method and an apparatus for
selectively reducing noise in a digital signal. In particular, the
method and apparatus are based on using integer arithmetic.
BACKGROUND
[0002] Wavelet thresholding is a sophisticated and effective
approach for noise reduction. Thereby, wavelet transformations
allow a finely graduated thresholding of the wavelet coefficients,
resulting in a finely graduated noise reduction, which in most
cases is the goal for noise reduction applications. Wavelet
transformations are usually calculated using floating-point
arithmetic.
[0003] For wavelet thresholding a wavelet decomposition of the
input data is performed. For digital input data, this is done with
the discrete wavelet transform (DWT), which can be realized with
filter banks. Thereby, each cascading step is called a level. The
DWT decomposes the input data into approximation and detail
coefficients. FIG. 1 shows a simple one dimensional DWT with a
low-pass filter L, a high-pass filter H and subsequent downsampling
stages.
[0004] A two-dimensional DWT can be realized with one-dimensional
DWTs by applying a row-column separation. Furthermore, the wavelet
decomposition can be iterated resulting into a multi-level
decomposition. FIG. 2 shows the wavelet decomposition for a 2D-DWT
resulting in four sub-bands LL, LH, HL, HH. Sub-band LL contains
the approximation coefficients, the other sub-bands detail
coefficients.
[0005] FIG. 3 depicts a multi-level decomposition for a wavelet
analysis based on a 2D-DWT, in which sub-band LL (i.e.
approximation coefficients) of a level i is the input for the next
level i+1. FIG. 3 shows three such levels.
[0006] Subsequent to the decomposition, the thresholding or
shrinking of the detail wavelet coefficients (for 2D-DWT: e.g.
LH.sub.i, HL.sub.i, HH.sub.i with i=1, . . . , N) is performed
depending on the assigned noise model. This--in principle--leads to
the reduction of noise. Thereby, the values of the wavelet
coefficients w are modified based on a given threshold value T into
resulting wavelet coefficients WT. There are certain established
thresholding techniques. Amongst others these are
hard-thresholding, soft-thresholding and other, more sophisticated
thresholding-techniques, e.g. Zhang (Zhang, X.-P. and Desai, M. D.:
Adaptive denoising based on SURE risk. In: IEEE Signal Processing
Letters, 1998, vol. 5, no. 10, pp. 265-267). The following
equations and FIG. 4 further describe the hard- and
soft-thresholding techniques.
w T , hard = { w , w > T 0 , w .ltoreq. T ( Eq . 1.1 ) w T ,
soft = { w - T , w .gtoreq. T w + T , w .ltoreq. - t 0 , w < T (
Eq . 1.2 ) ##EQU00001##
[0007] The straight-forward hard-thresholding benefits from the
precision that floating-point arithmetic provides, as a finely
graduated selection of the threshold value T is possible and
therefore it can precisely be defined which wavelet values are to
be set to zero. Soft-thresholding or further sophisticated
thresholding-techniques also prune or otherwise reduce the wavelet
coefficients, e.g. w-T. Consequently, with floating-point
arithmetic also a finely graduated pruning is possible.
[0008] The use of floating-point arithmetic requires the use of
architectures and/or processing platforms that support
floating-point arithmetic. This can be considered to be at least
expensive, and even inefficient for hardware (HW) processing
platforms like FGPAs or certain digital signal processors. In order
to circumvent this restriction, some wavelet transformations make
use of filter coefficients that allow a processing in fixed-point
arithmetic at a precision equal to floating-point arithmetic.
Fixed-point arithmetic in general requires a growing precision for
each level of the wavelet decomposition tree, which easily exceeds
the usually available 16 or 32 bit. Consequently, those methods are
also not efficient or even not applicable for those kinds of
platforms.
[0009] A further solution is the use of an integer wavelet
transform (IWT), which allows a mapping of integer input values to
integer output values. The processing in between needs not
necessarily be based on integer arithmetic. In addition, with the
selection of special filter coefficients that allow a processing
with integer arithmetic, the entire processing including the
thresholding can be performed in integer arithmetic. Thereby, with
the resulting integer wavelet coefficients a finely graduated
thresholding including pruning is not possible, as the finest
achievable graduation is the precision that integer arithmetic
provides. Consequently, the processing has to rely on a less finely
or even coarsely graduated thresholding, as e.g. a pruning in the
decimal places is not supported. The coarsely graduated
thresholding even remains if floating-point arithmetic is used for
the in-between processing of the IWT.
[0010] One example, which allows the fixed-point approach as well
as an IWT calculation completely with integer arithmetic, is the
lifting implementation of the 5-tap/3-tap filter by Le
Gall.sup.1,2. The equations for the lifting scheme for the one
dimensional 5-tap/3-tap filter are shown below, where y(2n) (Eq.
2.1) represents the output of the high-pass and y(2n+1) (Eq. 2.2)
the output of the low-pass. .sup.1 Le Gall, D. & Tabatabai, A.:
Sub-band coding of digital images using symmetric short kernel
filters and arithmetic coding techniques. In: International
Conference on Acoustics, Speech and Signal Processing, 1988, pp.
761-764 vol. 2.sup.2 Skodras, A. et al. The JPEG 2000 still image
compression standard. In: Signal Processing Magazine, IEEE, 2001,
vol. 18, no. 5, pp. 36-58
y ( 2 n + 1 ) = x ( 2 n + 1 ) - x ( 2 n ) + x ( 2 n + 2 ) 2 ( Eq .
2.1 ) y ( 2 n ) = x ( 2 n ) + y ( 2 n - 1 ) + y ( 2 n + 1 ) + 2 4 (
Eq . 2.2 ) ##EQU00002##
[0011] Taking the fixed-point approach above, for an input of 10
bit signed, the coefficients y(2n+1) are 12 bit signed and the
coefficients y(2n) are 14 bit signed. Respective word widths are
required for the processing steps of the calculation. With a simple
approximation, namely 4-bit width increase for the low-pass and
2-bit increase for the high-pass, the required precision for a
multi-level 2D-DWT can be estimated. For the first level, the
coefficients of the sub-band LL.sub.1 require a word width of 18
bits, for the second level the coefficients of sub-band LL.sub.2
require 26 bits and for the third level the coefficients of
sub-band LL.sub.3 require 34 bits.
SUMMARY OF THE INVENTION
[0012] A problem to be solved is how to achieve a finely graduated
wavelet thresholding using integer arithmetic with minimal
precision (with respect to the required word width, i.e. precision
of calculation), wherein overflow is to be prevented.
[0013] This problem is solved by the method disclosed in claim 1. A
corresponding apparatus that utilizes the method is disclosed in
claim 9.
[0014] The present invention applies a range expansion to the input
values before, during and/or after the transform steps, then the
thresholding and then an inverse expansion before, during and/or
after the transform steps, wherein the transform steps use integer
arithmetic. Thus, also the thresholding works on integer values,
and is therefore easier to implement than conventional
thresholding. In the thresholding, the integer coefficients are
compared with appropriate thresholds which may also be
integers.
[0015] Two basic embodiments are described that achieve a finely
graduated wavelet thresholding using integer arithmetic with
minimal processing precision.
[0016] According to one aspect of the present invention, a method
for selectively reducing noise in a digital signal having a first
range of values comprises the steps of decomposing the digital
signal to a plurality of frequency sub-bands and expanding the
digital signal or at least one of said frequency sub-bands by one
or more bits before, during or after the step of decomposing,
wherein at least one expanded frequency sub-band having a higher
range of integer values than the first range of integer values is
obtained,
reducing (in terms of absolute value) or removing in at least one
expanded frequency sub-band values that are below a threshold, and
reconstructing from the frequency sub-bands, after said step of
reducing or removing in the at least one expanded frequency
sub-band the values that are below a threshold, an output signal
having the first range of values, wherein the at least one expanded
frequency sub-band or the reconstructed output signal is
de-expanded.
[0017] According to another aspect of the present invention, an
apparatus for selectively reducing noise in a digital signal having
a first range of values comprises means for decomposing the digital
signal to a plurality of frequency sub-bands and means for
expanding the digital signal or at least one of said frequency
sub-bands by one or more bits before, during or after the
decomposing, wherein at least one expanded frequency sub-band
having a higher range of integer values than the first range of
integer values is obtained,
means for reducing or removing in at least one expanded frequency
sub-band values that are below a threshold, and means for
reconstructing from the frequency sub-bands, including the at least
one expanded frequency sub-band, an output signal having the first
range of values, wherein the apparatus comprises means for
de-expanding the at least one expanded frequency sub-band or the
reconstructed output signal.
[0018] In a first basic embodiment, a pre-processing step is
introduced in which the range of the input values is expanded to
the level of precision required for the thresholding. Thereby, the
values are distributed over the expanded value range.
[0019] In a second basic embodiment, a distributed range expansion
is applied during the processing. This can be obtained by omitting
the necessary division (by 2 and 4 in Eq. 2.1 and 2.2) for
fractional coefficients and adjusting the other coefficients
accordingly, thus selectively introducing fixed-point precision.
This is utilized in some or all stages until the required precision
is achieved. This expands the range of the resulting or
intermediate wavelet coefficients. With this range expansion
obtained within the pre-processing step or distributed over the
processing, a higher precision is achieved which allows for a
finely graduated thresholding. After the inverse transformation
this range expansion is undone by corresponding compression or
de-expansion.
[0020] Advantageous embodiments of the invention are disclosed in
the dependent claims, the following description and the
figures.
BRIEF DESCRIPTION OF THE DRAWINGS
[0021] Exemplary embodiments of the invention are described with
reference to the accompanying drawings, which show in
[0022] FIG. 1 a 1-dimensional DWT;
[0023] FIG. 2 a 2-dimensional DWT;
[0024] FIG. 3 multi-level decomposition for wavelet analysis using
2D-DWT;
[0025] FIG. 4 different thresholding techniques;
[0026] FIG. 5 a simple range expansion;
[0027] FIG. 6 a second range expansion technique;
[0028] FIG. 7 the decomposition and composition part of a wavelet
filter according to a first embodiment of the invention;
[0029] FIG. 8a the decomposition part of a wavelet filter according
to a second embodiment of the invention;
[0030] FIG. 8b the composition part of a wavelet filter according
to a second embodiment of the invention; and
[0031] FIG. 9 the RMSE between processed and unprocessed data set
using simple bit shift, depending on expansion width k and
threshold value T.
DETAILED DESCRIPTION OF THE INVENTION
[0032] While in the following the 5-tap/3-tap filter is taken as a
reference example, the invention applies also to other means for
decomposition (also called analysis) and means for reconstruction
(also called composition or synthesis).
[0033] FIG. 7 shows a wavelet filter bank based on multi-level
decomposition and composition, where the input values in.sub.orig
are expanded by an expand function Exp which results in expanded
input values in.sub.exp. The expanded input values in.sub.exp are
used for the decomposition stage 2D-DWT, which is a discrete
wavelet transform in this case. To the output values of each
sub-band of each level, thresholding functions Th.sub.LH1,
Th.sub.HL1, . . . , Th.sub.HH3 are applied to remove (or at least
reduce in terms of absolute value) coefficients that are below the
respective threshold. These represent usually mainly noise,
assuming that the thresholds are properly set. Thus, the noise is
cancelled. Also small coefficients coming from the original signal
are removed, but these are irrelevant since they cannot be
distinguished from the noise.
[0034] In this particular example, the thresholds used by the
thresholding blocks Th.sub.LH1, Th.sub.HL1, Th.sub.HH1 applied to
the first decomposition level may be all different, and may be
different from those of the thresholding blocks Th.sub.LH2,
Th.sub.HL2, Th.sub.HH2 applied to the second decomposition level
and those of the thresholding blocks Th.sub.LL3, Th.sub.LH3,
Th.sub.HL3, Th.sub.HH3 applied to the third decomposition level.
The thresholds may however be independent from each other, and may
thus be equal. They may be defined by any adequate method, e.g.
generalized cross validation.
[0035] After the expansion, wavelet decomposition and selective
thresholding, the signal is reconstructed from the resulting
sub-bands. The coefficients of the sub-bands are fed into an
inverse discrete wavelet transform (IDWT) 2D-IDWT, which is
complementary to the previously used DWT and therefore
2-dimensional in this case. Thus, a perfect reconstruction of the
input signal is generally possible (assuming that DWT and IDWT
blocks meet the well-known perfect reconstruction conditions),
except where spectral components below the respectively applied
thresholding have been removed.
[0036] In a final stage, the range of the IDWT output values
out.sub.exp is re-mapped to the original range of values by an
inverse expansion InvExp, resulting in reconstructed, de-expanded
output values out.sub.de-exp that contain reduced noise.
[0037] It is a particular advantage of the invention that all the
processing steps can be executed using integer arithmetic. Due to
the range expansion, the available value range for the processing
can be fully utilized even if the range of input values is smaller.
In particular, the range expansion can be selectively applied to
certain frequency bands of the signal, as further described below,
so that selective noise reduction can be performed.
[0038] It is clear that while in this example a 2-dimensional DWT
decomposition is used, it may also be 1-dim or 3-dim or any other
type of signal decomposition. Further, thresholding needs not be
applied to all sub-bands of any decomposition level, and not to all
decomposition levels. The thresholding may also be applied only to
frequency bands where noise is mainly expected, e.g. in FIG. 7
LH.sub.1, HL.sub.1, HH.sub.1, and HH.sub.2 as being the four most
detailed coefficients. Particularly in the lowest sub-band
LL.sub.3, thresholding is normally skipped. Generally, it depends
on the frequency spectrum of the signal and of the expected noise
in which sub-bands thresholding are used. In programmable
architectures, the thresholds can be individually set, and
thresholding can be skipped by setting the threshold to zero.
According to the invention, the values to which the thresholds are
applied are all integers. Thus, the thresholds themselves may, but
need not be integers.
[0039] The invention utilizes a range expansion applied to the
values to be processed, either prior to the processing or
dynamically (distributed) during the processing or both, in order
to allow a finely graduated thresholding of the wavelet
coefficients. Thereby, the required processing precision, i.e. the
finest graduation required, is adapted selectively to the available
processing architecture. The required precision depends on the
application that needs noise reduction, e.g. image processing.
[0040] In order to determine the minimum required precision, the
inherent range expansion, which can be caused by the
characteristics of the applied filters (e.g. the overshoot), has to
be taken into account. For the 5-tap/3-tap filter, this can be
roughly estimated to be 2 bit for the first three levels for signed
input values. For the example above with 10-bit signed input
values, the output coefficients of the third level require 12 bit
for the calculation of the IWT. Therefore, e.g. for a 16-bit
integer arithmetic 4 bit are available to increase the thresholding
precision. For 9-bit signed these are 5 bit and so on. For 24-bit
or 32-bit integer arithmetic there are more bits available to
increase the thresholding precision.
[0041] In the following, the number of bits used to increase the
thresholding precision is denoted with k. For embodiments where the
range expansion is performed prior to the processing, the range of
the input values having a width of n bit is expanded by k bit into
a resulting word width of k+n bit. Thereby, several expansion
schemes are possible. Two range expansion schemes are described in
the following.
[0042] A first scheme (proposed scheme 1) is based on a simple bit
shift, as shown in FIG. 5. In this case, k bits are appended that
all have the value 0. This may also be described as a
multiplication with 2.sup.k or a logical bit shift by k:
x.sub.S2=x*2.sup.k (Eq. 3.1)
x.sub.S2=x<<k (Eq. 3.2)
[0043] Obviously, the resulting expanded values are not evenly
distributed, since their k least significant bits (LSBs) are all
zero.
[0044] This range expansion can be inverted by a logical bit shift
by -k, truncating the k LSBs and rounding the result to the nearest
integer. E.g. if a value of 010001.sub.bin is to be de-expanded by
2 bits, the intermediate result is 0100.01.sub.bin which is rounded
to 0100.sub.bin, whereas for a value of 010010.sub.bin the
intermediate result is 0100.10 which is rounded to
0101.sub.bin.
[0045] This first scheme is very easy to implement, and can be
executed very fast.
[0046] A second scheme (proposed scheme 2) allows for a more even
distribution of the original input values over the new range of n+k
bit. This scheme is particularly advantageous for unsigned integer
input values, which is the case e.g. in image processing. Provided
that k<=n, the k most significant bits (MSBs) of the n original
bits are inserted at the LSB position after the shift of k bits.
FIG. 6 depicts this second scheme.
[0047] E.g. if values of 0100101.sub.bin, 0110110.sub.bin and
0001010.sub.bin are to be expanded by 3 bits, the results are
0100101010.sub.bin, 0110110011.sub.bin and 0001010000.sub.bin since
the 3 MSBs are appended (as underlined). This scheme has the
advantage that the resulting expanded values are more evenly
distributed. For signed integer input values the processing is in
principle the same, however in the case of a signed value the MSB
may be omitted in order to achieve different values.
[0048] This scheme can be easily implemented in hardware. In the
case that k>n, the MSBs of the n bit can be replicated until n+m
is equal to k. E.g. expansion of the 4-bit value 1011.sub.bin by 6
bits results in 1011101110.sub.bin. Moreover, this proposed scheme
2 can be universally applied for achieving a more even distribution
in the expansion of data sets to a higher data range. For the
de-expansion, the respective MSBs (irrespective of their particular
value) that were added during expansion need to be subtracted. E.g.
if the two MSBs of an input value were added during expansion, then
the two MSBs of the resulting value need to be subtracted during
the de-expansion.
[0049] In the following, the above-described schemes are compared
to three conventional schemes for the case k<=n with k=4 and
n=8. The reference is an exact even distribution, which however
would require floating-point arithmetic:
x ex = x * 4095 255 ( Eq . 4 ) ##EQU00003##
[0050] The first conventional scheme Ref1 uses rounding, the second
conventional scheme Ref2 the floor-function and the third
conventional scheme Ref3 the ceiling-function:
x ex , 1 = ROUND ( x ex ) x ex , 2 = x ex x ex , 3 = x ex ( Eq . 5
) ##EQU00004##
[0051] Tab. 1 lists the results of this comparison, in particular
the achieved errors and the mean square errors (MSE).
TABLE-US-00001 TABLE 1 Comparison of different expansion schemes
Proposed Proposed X.sub.ex = X* Ref1 Ref2 Ref3 Scheme 1 Scheme 2 X
4095/255 Error Error Error Error Error 0 0.00 0 0.00 0 0.00 0 0.00
0 0.00 0 0.00 1 16.06 16 -0.06 16 -0.06 17 0.94 16 -0.06 16 -0.06 2
32.12 32 -0.12 32 -0.12 33 0.88 32 -0.12 32 -0.12 3 48.18 48 -0.18
48 -0.18 49 0.82 48 -0.18 48 -0.18 4 64.24 64 -0.24 64 -0.24 65
0.76 64 -0.24 64 -0.24 5 80.29 80 -0.29 80 -0.29 81 0.71 80 -0.29
80 -0.29 6 96.35 96 -0.35 96 -0.35 97 0.65 96 -0.35 96 -0.35 7
112.41 112 -0.41 112 -0.41 113 0.59 112 -0.41 112 -0.41 8 128.47
128 -0.47 128 -0.47 129 0.53 128 -0.47 128 -0.47 9 144.53 145 0.47
144 -0.53 145 0.47 144 -0.53 144 -0.53 10 160.59 161 0.41 160 -0.59
161 0.41 160 -0.59 160 -0.59 11 176.65 177 0.35 176 -0.65 177 0.35
176 -0.65 176 -0.65 12 192.71 193 0.29 192 -0.71 193 0.29 192 -0.71
192 -0.71 13 208.76 209 0.24 208 -0.76 209 0.24 208 -0.76 208 -0.76
14 224.82 225 0.18 224 -0.82 225 0.18 224 -0.82 224 -0.82 15 240.88
241 0.12 240 -0.88 241 0.12 240 -0.88 240 -0.88 16 256.94 257 0.06
256 -0.94 257 0.06 256 -0.94 257 0.06 17 273.00 273 0.00 273 0.00
273 0.00 272 -1.00 273 0.00 . . . 255 4095.00 4095 0 4095 0 4095 0
4080 -15.00 4095 0.00 Min. Error -0.47 -0.94 0.00 -15.00 -0.88 Max.
Error 0.47 0.00 0.94 0.00 0.88 MSE 0.083 0.303 0.303 75.147
0.147
[0052] E.g. for an input value of 16 that is expanded from 8 to 12
bit, a floating-point arithmetic would calculate with a value of
256.94 and all the integer methods would use 256 (where the error
is -0.94) or 257 (where the error is 0.06) instead. However, the
MSE is an important key factor and differs for the shown expansion
schemes.
[0053] The proposed scheme 2 proofs to be superior to scheme Ref2
and Ref3 and comes close to scheme Ref1, which is more costly to
implement. The proposed scheme 1, which achieves a less even
distribution, has the advantage that it allows a rather easy
implementation.
[0054] While the described first approach expands the values before
decomposition and de-expands or compresses the values again after
reconstruction, an alternative second approach is available, as
already mentioned.
[0055] The second approach makes use of a distributed (or dynamic)
range expansion. Thereby, the necessary division while calculating
the coefficients are suppressed and thus fixed-point arithmetic to
a certain level is introduced. For the lifting implementation of
the 5-tap/3-tap filter, this leads to the following equations (cf.
Eq. 2):
y(2n+1)=2*x(2n+1)-(x(2n)+x(2n+2)) (Eq.6.1)
y(2n)=8*x(2n)+y(2n-1)+y(2n+1) (Eq. 6.2)
[0056] Thereby, the range of the coefficients y(2n+1) is expanded
by 1 bit and the range of the coefficients y(2n) by three bits in
the decimal places. This second approach can be applied for each
1-dim DWT. Consequently, for each one-level 2D-DWT it can be
selected whether the expansion for sub-band LL, which is the
relevant sub-band for a multi-level wavelet analysis, is 6, 3 or 0
bit. For the sub-band HL this expansion is 4, 1 or 0 bit and for
sub-band LH 4, 3 or 0 bit. Finally, the expansion of sub-band HH is
2, 1 or 0 bit. In contrast to the second embodiment, the range
expansion can only be done in quantized steps. If the required
range expansion is reached, e.g. 16 bit in the final inner stage,
the enforced distributed expansion is simply turned off, i.e.
subsequent decomposition stages include the division step again.
This means that, although small range extension is still possible
due to the filter architecture, this is already considered so that
all processing is done with not more than the required range
expansion, e.g. 16 bit.
[0057] It has again to be noted that instead of the original input
values, the expanded input values are transformed. The resulting
coefficients are thresholded and then the inverse transformation is
performed. Finally, the introduced range expansion is undone by
de-expanding (or compressing) the results.
[0058] FIG. 8 shows an embodiment where the range expansion is
distributed. Instead of the common DWT stages of FIG. 7, modified
DWT stages 2D-DWT' are used that apply Eq.6 to the input values.
The value expansion is done by omitting the division operation of
Eq.2. Again, the thresholding Th'.sub.LH1, Th'.sub.HL1, . . . ,
Th'.sub.HH3 is done on the expanded values. The decomposition is
reversed by reconstruction stages 2D-IDWT' that are complementary
to the decomposition stages. Thus, a perfect reconstruction of the
input signal is possible if all thresholds are set to zero. Thus
the frequency spectrum of the output signal is not modified,
compared to the input signal. However, some or all sub-bands or
wavelets can be expanded in order to enable a finely graduated
wavelet thresholding.
[0059] For the 5-tap/3-tap filter, experiments show that with the
range expansion a finely graduated wavelet thresholding is possible
using integer arithmetic with minimal processing precision (or
rather maximum usage of the available processing precision by value
range expansion). With each expansion bit added, the wavelet
thresholding becomes more finely graduated. Thereby, next to the
maximum difference that can be observed between the original and
the processed data set (e.g. image) it is important how many data
items (e.g. pixels) are affected. The latter can be expressed e.g.
by the root mean square error (RMSE) between processed and
unprocessed data set. FIG. 9 shows the results for the first
approach (i.e. bit shift and fill with zeros) with proposed
expansion scheme 1 depending on the expansion width k and the
threshold value T. The input data width is 9 bit unsigned. The
absolute maximum difference max(|d|) and the RMSE
( 1 N N d 2 ) ##EQU00005##
are depicted.
[0060] The key advantages of the invention are the use of integer
arithmetic for wavelet thresholding which allows the use of a wider
class of architectures and processing platforms. Furthermore, the
precision, i.e. the word width, for the processing can be minimized
in comparison to fixed-point arithmetic. Both advantages, integer
arithmetic and minimized precision requirements, can result into
lower costs with an equal or similar thresholding quality. They
allow a more efficient implementation and therefore the selection
of smaller devices, or devices with less processing
performance.
[0061] It will be understood that the present invention has been
described purely by way of example, and modifications of detail can
be made without departing from the scope of the invention. Each
feature disclosed in the description and (where appropriate) the
claims and drawings may be provided independently or in any
appropriate combination. Features may, where appropriate be
implemented in hardware, software, or a combination of the two.
[0062] Connections may, where applicable, be implemented as
wireless connections or wired, not necessarily direct or dedicated,
connections.
[0063] Reference signs appearing in the claims are by way of
illustration only and shall have no limiting effect on the scope of
the claims.
* * * * *